scholarly journals A New Criterion for Affineness

ISRN Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Jing Zhang
Keyword(s):  
Characteristic Zero ◽  
Large Degree ◽  
Direct Application ◽  
Closed Field ◽  
Affine Variety ◽  
Algebraically Closed Field ◽  
New Criterion

We show that an irreducible quasiprojective variety of dimension defined over an algebraically closed field with characteristic zero is an affine variety if and only if () = 0 and () = 0 for all , , where is any hypersurface with sufficiently large degree. A direct application is that an irreducible quasiprojective variety over is a Stein variety if it satisfies the two vanishing conditions. Here, all sheaves are algebraic.

scholarly journals Commutative algebraic monoid structures on affine surfaces

2020 ◽  
Vol 33 (1) ◽  
pp. 177-191
Author(s):  
Sergey Dzhunusov ◽  
Yulia Zaitseva
Keyword(s):  
Characteristic Zero ◽  
Closed Field ◽  
Affine Variety ◽  
Commutative Monoid ◽  
General Classification ◽  
Algebraically Closed Field ◽  
Algebraic Monoid

Abstract We classify commutative algebraic monoid structures on normal affine surfaces over an algebraically closed field of characteristic zero. The answer is given in two languages: comultiplications and Cox coordinates. The result follows from a more general classification of commutative monoid structures of rank 0, n - 1 n-1 or 𝑛 on a normal affine variety of dimension 𝑛.

scholarly journals An algebraically closed field

1968 ◽  
Vol 9 (2) ◽  
pp. 146-151 ◽  
Author(s):  
F. J. Rayner
Keyword(s):  
Positive Integer ◽  
Power Series ◽  
Formal Power Series ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Zero Characteristic ◽  
Image Position ◽  
Formal Power ◽  
Puiseux Expansions

Letkbe any algebraically closed field, and denote byk((t)) the field of formal power series in one indeterminatetoverk. Letso thatKis the field of Puiseux expansions with coefficients ink(each element ofKis a formal power series intl/rfor some positive integerr). It is well-known thatKis algebraically closed if and only ifkis of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous toKwhich is algebraically closed whenkhas non-zero characteristicp. In this paper, I prove that the setLof all formal power series of the form Σaitei(where (ei) is well-ordered,ei=mi|nprt,n∈ Ζ,mi∈ Ζ,ai∈k,ri∈ Ν) forms an algebraically closed field.

scholarly journals On the eigenvalues of some class of pseudo-linear transformations

2007 ◽  
pp. 79-83
Author(s):  
Milica Andjelic
Keyword(s):  
Linear Transformation ◽  
Closed Field ◽  
Linear Transformations ◽  
Algebraically Closed Field ◽  
New Criterion

We develop a connection between the eigenvalues of a class of pseudo-linear transformation over a field K and the eigenvalues of a certain linear transformation. We give a new criterion for this class to be diagonalizable over algebraically closed field.

scholarly journals Fibrations with moving cuspidal singularities

1991 ◽  
Vol 122 ◽  
pp. 161-179 ◽  
Author(s):  
Yoshifumi Takeda
Keyword(s):  
Algebraic Geometry ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Closed Field ◽  
Projective Surface ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Smooth Projective Surface ◽  
Almost All

Let f: V → C be a fibration from a smooth projective surface onto a smooth projective curve over an algebraically closed field k. In the case of characteristic zero, almost all fibres of f are nonsingular. In the case of positive characteristic, it is, however, known that there exist fibrations whose general fibres have singularities. Moreover, it seems that such fibrations often have pathological phenomena of algebraic geometry in positive characteristic (see M. Raynaud [7], W. Lang [4]).

Abelian Gradings on Upper Block Triangular Matrices

2012 ◽  
Vol 55 (1) ◽  
pp. 208-213 ◽  
Author(s):  
Angela Valenti ◽  
Mikhail Zaicev
Keyword(s):  
Abelian Group ◽  
Characteristic Zero ◽  
Finite Abelian Group ◽  
Triangular Matrix ◽  
Closed Field ◽  
Triangular Matrices ◽  
Algebraically Closed Field ◽  
Matrix Algebras ◽  
Block Triangular Matrix

AbstractLet G be an arbitrary finite abelian group. We describe all possible G-gradings on upper block triangular matrix algebras over an algebraically closed field of characteristic zero.

scholarly journals Resolutions With Conical Slices and Descent for the Brauer Group Classes of Certain Central Reductions of Differential Operators in Characteristic p

2019 ◽  
Author(s):  
Dmitry Kubrak ◽  
Roman Travkin
Keyword(s):  
Differential Operators ◽  
Closed Field ◽  
Brauer Group ◽  
Affine Variety ◽  
Smooth Variety ◽  
Characteristic P ◽  
Algebraically Closed Field ◽  
Quiver Varieties ◽  
Hypertoric Varieties ◽  
General Reduction

Abstract “Even more so is the word ‘crystalline’, a glacial and impersonal concept of his which disdains viewing existence from a single portion of time and space” Eileen Myles, “The Importance of Being Iceland” For a smooth variety $X$ over an algebraically closed field of characteristic $p$ to a differential 1-form $\alpha $ on the Frobenius twist $X^{\textrm{(1)}}$ one can associate an Azumaya algebra ${{\mathcal{D}}}_{X,\alpha }$, defined as a certain central reduction of the algebra ${{\mathcal{D}}}_X$ of “crystalline differential operators” on $X$. For a resolution of singularities $\pi :X\to Y$ of an affine variety $Y$, we study for which $\alpha $ the class $[{{\mathcal{D}}}_{X,\alpha }]$ in the Brauer group $\textrm{Br}(X^{\textrm{(1)}})$ descends to $Y^{\textrm{(1)}}$. In the case when $X$ is symplectic, this question is related to Fedosov quantizations in characteristic $p$ and the construction of noncommutative resolutions of $Y$. We prove that the classes $[{{\mathcal{D}}}_{X,\alpha }]$ descend étale locally for all $\alpha $ if ${{\mathcal{O}}}_Y\widetilde{\rightarrow }\pi _\ast{{\mathcal{O}}}_X$ and $R^{1}\pi _*\mathcal O_X = R^2\pi _*\mathcal O_X =0$. We also define a certain class of resolutions, which we call resolutions with conical slices, and prove that for a general reduction of a resolution with conical slices in characteristic $0$ to an algebraically closed field of characteristic $p$ classes $[{{\mathcal{D}}}_{X,\alpha }]$ descend to $Y^{\textrm{(1)}}$ globally for all $\alpha $. Finally we give some examples; in particular, we show that Slodowy slices, Nakajima quiver varieties, and hypertoric varieties are resolutions with conical slices.

scholarly journals Involutions on finite-dimensional algebras over real closed fields

2004 ◽  
Vol 77 (1) ◽  
pp. 123-128 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Characteristic Zero ◽  
Real Closed Field ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
Real Closed Fields ◽  
Finite Dimensional ◽  
Closed Fields ◽  
Finite Dimensional Algebras

AbstractIt is shown that the following conditions on a finite-dimensional algebra A over a real closed field or an algebraically closed field of characteristic zero are equivalent: (i) A admits a special involution, in the sense of Easdown and Munn, (ii) A admits a proper involution, (iii) A is semisimple.

scholarly journals Corrections to “Seminormal rings and weakly normal varieties”

1987 ◽  
Vol 107 ◽  
pp. 147-157 ◽  
Author(s):  
Marie A. Vitulli
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Regular Function ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Regular Functions ◽  
Normal Varieties

In “Seminormal rings and weakly normal varieties” we introduced the notion of a c-regular function on an algebraic variety defined over an algebraically closed field of characteristic zero. Our intention was to describe those k-valued functions on a variety X that become regular functions when lifted to the normalization of X, but without any reference to the normalization in the definition. That is, we aspired to identify the c-regular functions on X with the regular functions on the weak normalization of X

HOPF ALGEBRAS OF DIMENSION 30

2010 ◽  
Vol 09 (01) ◽  
pp. 11-15 ◽  
Author(s):  
DAIJIRO FUKUDA
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional

This paper contributes to the classification of finite dimensional Hopf algebras. It is shown that every Hopf algebra of dimension 30 over an algebraically closed field of characteristic zero is semisimple and thus isomorphic to a group algebra or the dual of a group algebra.

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